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Nous montrons que l’ensemble des racines modulo une puissance d’un nombre premier d’un polynôme à coefficients entiers de degré est une union d’au plus progressions arithmétiques de modules assez grands. Nous en déduisons une majoration du nombre de ses racines dans un intervalle réel court.
It is proved that the solution to the initial value problem , u(0,x) = 1/(1+x²), does not belong to the Gevrey class in time for 0 ≤ s < 1. The proof is based on an estimation of a double sum of products of binomial coefficients.
Let , and be fixed complex numbers. Let be the Toeplitz matrix all of whose entries above the diagonal are , all of whose entries below the diagonal are , and all of whose entries on the diagonal are . For , each principal minor of has the same value. We find explicit and recursive formulae for the principal minors and the characteristic polynomial of . We also show that all complex polynomials in are Toeplitz matrices. In particular, the inverse of is a Toeplitz matrix when...
A slight modification of the proof of Szemerédi’s cube lemma gives that if a set satisfies , then must contain a non-degenerate Hilbert cube of dimension . In this paper we prove that in a random set determined by for , the maximal dimension of non-degenerate Hilbert cubes is a.e. nearly and determine the threshold function for a non-degenerate -cube.
Let be an odd prime number with q an odd integer. Let δ (resp. φ) be an odd (resp. even) Dirichlet character of conductor p and order (resp. order dividing q), and let ψₙ be an even character of conductor and order pⁿ. We put χ = δφψₙ, whose value is contained in . It is well known that the Bernoulli number is not zero, which is shown in an analytic way. In the extreme cases and q, we show, in an algebraic and elementary manner, a stronger nonvanishing result: for any pⁿth root ξ...
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